Subgroups — #26233040

Order 12032.

1314 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#10625329	C₂	1	2	yes	maximal, complemented
2	#20244229	C₂	1	6	yes	maximal, complemented
3	#20243380	C₂	1	3	yes	maximal, complemented
4	C₄×#13093941	C₂	1	3	yes	maximal, complemented
5	C₂×C₄₇×#550	C₂	1	1	yes	maximal, characteristic
6	#15689254	—	2	6	no	—
7	Q₈×C₄₇⋊C₈	—	2	2	no	—
8	#15689147	C₂²	1	6	yes	—
9	C₂×#13093941	C₂²	1	6	yes	—
10	C₂×#13093941	C₂²	1	6	yes	—
11	C₄×#10625329	C₂²	1	6	yes	—
12	#15688892	C₂²	1	3	yes	—
13	#15688892	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₄₇	C₄	1	2	yes	—
15	C₄₇×#550	—	2	2	no	—
16	C₄₇×#550	—	2	6	no	—
17	C₂×#127×C₄₇	C₂²	1	3	yes	—
18	C₂×#121×C₄₇	C₄	1	6	yes	—
19	C₂×C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₄₇	C₂²	1	3	yes	—
21	C₂×#10625329	C₂³	1	6	yes	—
22	C₂×#10625329	C₂³	1	2	yes	—
23	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	6	no	—
25	#13093941	—	2	12	no	—
26	#13093941	—	4	6	no	—
27	#13093935	—	2	6	no	—
28	C₄×C₄₇⋊C₈	—	2	6	no	—
29	C₂×#41×C₄₇	—	2	6	no	—
30	C₂×#41×C₄₇	—	2	2	no	—
31	C₂×#41×C₄₇	—	4	8	no	—
32	C₂²×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₄×C₄₇	C₂×C₄	1	6	yes	—
34	C₂²×D₄×C₄₇	C₂×C₄	1	6	yes	—
35	C₂²×D₄×C₄₇	C₂³	1	3	yes	—
36	C₂³×C₄×C₄₇	C₂×C₄	1	6	yes	—
37	#127×C₄₇	—	2	6	no	—
38	#127×C₄₇	—	2	6	no	—
39	#121×C₄₇	—	2	12	no	—
40	#121×C₄₇	—	2	12	no	—
41	C₄×Q₈×C₄₇	D₄	1	2	yes	—
42	C₄×Q₈×C₄₇	D₄	1	6	yes	—
43	C₄×D₄×C₄₇	—	2	6	no	—
44	C₄×D₄×C₄₇	—	2	6	no	—
45	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	3	yes	—
46	C₂×C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
48	C₂×C₄²×C₄₇	C₂³	1	3	yes	—
49	C₂⁴×C₄₇	C₂²×C₄	1	6	yes	—
50	#41×C₄₇	—	4	24	no	—
51	#41×C₄₇	—	4	8	no	—
52	C₂×Q₈×C₄₇	—	2	4	no	—
53	C₂×Q₈×C₄₇	C₂²⋊C₄	1	3	yes	—
54	C₂×Q₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×D₄×C₄₇	—	2	6	no	—
56	C₂×D₄×C₄₇	—	2	6	no	—
57	C₂×D₄×C₄₇	—	2	12	no	—
58	C₂×D₄×C₄₇	—	2	6	no	—
59	C₂×D₄×C₄₇	—	2	6	no	—
60	C₂×D₄×C₄₇	—	2	3	no	—
61	C₂×D₄×C₄₇	—	2	3	no	—
62	C₂×D₄×C₄₇	—	4	12	no	—
63	C₂×D₄×C₄₇	—	8	6	no	—
64	C₂²×C₄×C₄₇	—	2	6	no	—
65	C₂²×C₄×C₄₇	—	2	6	no	—
66	C₂²×C₄×C₄₇	—	4	12	no	—
67	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₄₇	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₄₇	C₂²×C₄	1	3	yes	—
70	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
71	C₄⋊C₄×C₄₇	—	2	2	no	—
72	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
73	C₄⋊C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
74	C₂²⋊C₄×C₄₇	—	2	12	no	—
75	C₄²×C₄₇	C₂×D₄	1	6	yes	—
76	C₄²×C₄₇	C₂×D₄	1	6	yes	—
77	#10625329	—	2	2	no	—
78	#10625329	—	2	6	no	—
79	#10625329	—	2	6	no	—
80	#10625329	#41	1	4	yes	complemented
81	C₂×C₄₇⋊C₈	—	2	2	no	—
82	C₂×C₄₇⋊C₈	—	2	6	no	—
83	C₂³×C₄₇	—	2	6	no	—
84	C₂³×C₄₇	—	2	6	no	—
85	C₂³×C₄₇	—	2	6	no	—
86	C₂³×C₄₇	—	4	6	no	—
87	C₂³×C₄₇	—	4	6	no	—
88	C₂³×C₄₇	C₂³×C₄	1	1	yes	characteristic
89	Q₈×C₄₇	—	2	6	no	—
90	Q₈×C₄₇	#100	1	4	yes	complemented
91	D₄×C₄₇	—	4	12	no	—
92	D₄×C₄₇	—	4	6	no	—
93	D₄×C₄₇	—	4	6	no	—
94	D₄×C₄₇	—	4	12	no	—
95	D₄×C₄₇	—	8	12	no	—
96	C₂×C₄×C₄₇	—	2	6	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	4	3	no	—
99	C₂×C₄×C₄₇	—	4	12	no	—
100	C₂×C₄×C₄₇	—	4	12	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	6	yes	—
103	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
104	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
105	C₂×C₄×C₄₇	C₂²×D₄	1	2	yes	—
106	C₂×C₄×C₄₇	C₂×#41	1	2	yes	—
107	C₂×C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
109	C₄₇⋊C₈	—	2	4	no	—
110	C₄₇⋊C₈	—	4	6	no	—
111	#22536	—	47	1	no	maximal, Sylow
112	C₂²×C₄₇	—	2	1	no	—
113	C₂²×C₄₇	—	2	1	no	—
114	C₂²×C₄₇	—	4	6	no	—
115	C₂²×C₄₇	—	4	6	no	—
116	C₂²×C₄₇	—	4	6	no	—
117	C₂²×C₄₇	—	8	6	no	—
118	C₂²×C₄₇	—	8	6	no	—
119	C₂²×C₄₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
120	C₂²×C₄₇	C₄×#41	1	1	yes	characteristic
121	C₂²×C₄₇	#520	1	1	yes	characteristic
122	C₄×C₄₇	—	2	3	no	—
123	C₄×C₄₇	—	2	6	no	—
124	C₄×C₄₇	—	4	2	no	—
125	C₄×C₄₇	C₂×#100	1	6	yes	—
126	C₄×C₄₇	#548	1	2	yes	—
127	C₄×C₄₇	#550	1	2	yes	—
128	C₂×#550	D₄₇	1	1	yes	characteristic
129	Q₈×M₁₆	—	47	2	no	—
130	#1965	—	47	6	no	—
131	#1800	—	47	3	no	—
132	C₄×#100	—	47	3	no	—
133	C₂×C₄₇	—	2	1	no	—
134	C₂×C₄₇	—	2	1	no	—
135	C₂×C₄₇	—	8	6	no	—
136	C₂×C₄₇	#2341	1	1	yes	characteristic
137	C₂×C₄₇	C₂²×#100	1	1	yes	characteristic
138	C₂×C₄₇	#2888	1	1	yes	characteristic
139	C₂²×#41	Dic₄₇	1	2	yes	—
140	#550	—	2	2	no	—
141	#550	—	2	6	no	—
142	C₂×#127	C₂×D₄₇	1	3	yes	—
143	C₂×#121	Dic₄₇	1	6	yes	—
144	C₂×C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
145	C₂×C₄×D₄	C₂×D₄₇	1	3	yes	—
146	#446	—	94	6	no	—
147	C₈×Q₈	—	94	2	no	—
148	#423	—	47	6	no	—
149	C₂×#100	—	47	6	no	—
150	C₂×#100	—	47	6	no	—
151	C₄×M₁₆	—	47	6	no	—
152	#343	—	47	3	no	—
153	#343	—	47	1	no	—
154	C₄₇	#22536	1	1	yes	characteristic, complemented, Sylow
155	C₂×#41	—	2	6	no	—
156	C₂×#41	—	2	2	no	—
157	C₂×#41	—	4	8	no	—
158	C₂²×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
159	C₂²×D₄	C₂×Dic₄₇	1	6	yes	—
160	C₂²×D₄	C₂×Dic₄₇	1	6	yes	—
161	C₂²×D₄	C₂²×D₄₇	1	3	yes	—
162	C₂³×C₄	C₂×Dic₄₇	1	6	yes	—
163	C₂×M₁₆	—	47	2	no	—
164	C₂×M₁₆	—	47	6	no	—
165	#127	—	2	6	no	—
166	#127	—	2	6	no	—
167	#121	—	2	12	no	—
168	#121	—	2	12	no	—
169	C₄×Q₈	C₂²⋊D₄₇	1	2	yes	—
170	C₄×Q₈	C₂²⋊D₄₇	1	6	yes	—
171	C₄×D₄	—	2	6	no	—
172	C₄×D₄	—	2	6	no	—
173	C₂×C₄⋊C₄	C₂×Dic₄₇	1	3	yes	—
174	C₂×C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
175	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	6	yes	—
176	C₂×C₄²	C₂²×D₄₇	1	3	yes	—
177	C₄⋊C₈	—	94	6	no	—
178	C₄⋊C₈	—	94	6	no	—
179	#100	—	94	12	no	—
180	#100	—	188	6	no	—
181	#97	—	94	6	no	—
182	C₄×C₈	—	94	6	no	—
183	C₂⁴	C₂²×Dic₄₇	1	6	yes	—
184	#41	—	4	24	no	—
185	#41	—	4	8	no	—
186	C₂×Q₈	—	2	4	no	—
187	C₂×Q₈	C₂²⋊Dic₄₇	1	3	yes	—
188	C₂×Q₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
189	C₂×D₄	—	2	6	no	—
190	C₂×D₄	—	2	6	no	—
191	C₂×D₄	—	2	12	no	—
192	C₂×D₄	—	2	6	no	—
193	C₂×D₄	—	2	6	no	—
194	C₂×D₄	—	2	3	no	—
195	C₂×D₄	—	2	3	no	—
196	C₂×D₄	—	4	12	no	—
197	C₂×D₄	—	8	6	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	2	6	no	—
200	C₂²×C₄	—	4	12	no	—
201	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
202	C₂²×C₄	C₂²×Dic₄₇	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₄₇	1	3	yes	—
204	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
205	M₁₆	—	47	4	no	—
206	M₁₆	—	94	2	no	—
207	M₁₆	—	94	6	no	—
208	M₁₆	—	94	6	no	—
209	C₂×C₈	—	94	2	no	—
210	C₂×C₈	—	94	6	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	C₂²⋊Dic₄₇	1	6	yes	—
213	C₄⋊C₄	C₂²⋊Dic₄₇	1	6	yes	—
214	C₂²⋊C₄	—	2	12	no	—
215	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
216	C₄²	C₂×C₂²⋊D₄₇	1	6	yes	—
217	C₂³	—	2	6	no	—
218	C₂³	—	2	6	no	—
219	C₂³	—	2	6	no	—
220	C₂³	—	4	6	no	—
221	C₂³	—	4	6	no	—
222	C₂³	C₂³×Dic₄₇	1	1	yes	characteristic
223	Q₈	—	2	6	no	—
224	Q₈	#13093941	1	4	yes	complemented
225	D₄	—	4	12	no	—
226	D₄	—	4	6	no	—
227	D₄	—	4	6	no	—
228	D₄	—	4	12	no	—
229	D₄	—	8	12	no	—
230	C₂×C₄	—	2	6	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	3	no	—
233	C₂×C₄	—	4	12	no	—
234	C₂×C₄	—	4	12	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	6	yes	—
237	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
238	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	3	yes	—
239	C₂×C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
240	C₂×C₄	#41×D₄₇	1	2	yes	—
241	C₂×C₄	#13094070	1	1	yes	characteristic
242	C₂×C₄	#13094071	1	1	yes	characteristic
243	C₈	—	94	4	no	—
244	C₈	—	188	6	no	—
245	C₂²	—	2	1	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	4	6	no	—
248	C₂²	—	4	6	no	—
249	C₂²	—	4	6	no	—
250	C₂²	—	8	6	no	—
251	C₂²	—	8	6	no	—
252	C₂²	C₂²×C₂²⋊Dic₄₇	1	1	yes	characteristic
253	C₂²	#41×Dic₄₇	1	1	yes	characteristic
254	C₂²	#15689606	1	1	yes	characteristic
255	C₄	—	2	3	no	—
256	C₄	—	2	6	no	—
257	C₄	—	4	2	no	—
258	C₄	C₂×#13093941	1	6	yes	—
259	C₄	#15689780	1	2	yes	—
260	C₄	#15689792	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	8	6	no	—
264	C₂	#20246229	1	1	yes	characteristic
265	C₂	C₂²×#13093941	1	1	yes	characteristic
266	C₂	#20251867	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting, prime radical	C₂×C₄₇×#550	C₂
socle	C₂²×C₄₇	C₂²×C₂²⋊C₄
cosocle	C₂²×C₄×C₄₇	C₂⁴
hypercenter	C₂×#550	D₄₇
hypocenter	C₄₇	#22536
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×#550

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. Q₈×#10625329	C₂	semidirect
2. #20244229	C₂	semidirect
3. #20243380	C₂	semidirect
4. C₄×#13093941	C₂	semidirect
80. #10625329	#41	semidirect
90. Q₈×C₄₇	#100	semidirect
154. C₄₇	#22536	semidirect
224. Q₈	#13093941	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×#10625329	2
#20244229	2
#20243380	2
C₄×#13093941	2
C₂×C₄₇×#550	2
#22536	47